Asymptotic equivalence between sequence-divisor sets and irregular-prime sets

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Let QH(x)\mathcal Q_H(x), QH−(x)\mathcal Q_{H^-}(x), and QH+(x)\mathcal Q_{H^+}(x) count the prime divisors up to xx of the HH-, H−H^--, and H+H^+-sequences, respectively. Let PG(x)\mathcal P_G(x), PH−(x)\mathcal P_{H^-}(x), and PH+(x)\mathcal P_{H^+}(x) count the corresponding irregular primes. Assume that the associated Wieferich sets are o(x/log⁡x)o(x/\log x). Sequence-divisor asymptotic conjecture. Asymptotically,

QH(x)∼PG(x),QH−(x)∼PH−(x),QH+(x)∼PH+(x).\mathcal Q_H(x)\sim\mathcal P_G(x),\qquad \mathcal Q_{H^-}(x)\sim\mathcal P_{H^-}(x),\qquad \mathcal Q_{H^+}(x)\sim\mathcal P_{H^+}(x).

Here the asymptotics of the irregular-prime counting functions are supplied by the paper's preceding conjectures. The assertion reduces the divisor-counting problem to irregular-prime counting once the Wieferich sets are negligible; this negligibility itself is assumed rather than known.

References

Primary source

Pieter Moree and Pietro Sgobba, “Prime divisors of -Genocchi numbers and the ubiquity of Ramanujan-style congruences of level”, arXiv:2209.08047 (2022).

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